Operational Symmetry and the Structural Reduction of Quantum State Space

We prove a structural reduction theorem for finite-dimensional quantum state space under operational symmetry. Unitary conjugation induces an equivalence relation on the density-operator space, and we show that the resulting quotient space is naturally isomorphic to the classical spectral simplex. This identification is established via the factorization of admissible state functionals through unitary orbits and the universal property of quotient spaces. The reduction separates geometric structure from functional selection. Classical symmetry and compositional consistency conditions imposed on the reduced simplex uniquely determine the logarithmic entropy form. Consequently, the von Neumann entropy appears as a corollary of structural reduction combined with classical functional characterization. Rather than postulating unitary invariance as an axiom of entropy functionals, the present analysis shows that it arises as a structural consequence of representational redundancy in quantum state space. The analysis is restricted to finite-dimensional systems and does not invoke thermodynamic limits, dynamical considerations, or information-processing tasks. The results clarify the geometric origin of entropy uniqueness within the density-operator framework.